Subgroups — #465

Order 64.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#121	C₂	1	1	yes	maximal, characteristic
5	#106	C₂	1	1	yes	maximal, characteristic, complemented
6	#103	C₂	1	1	yes	maximal, characteristic, complemented
7	#102	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈	C₂²	1	1	yes	characteristic
9	C₂×D₄	—	2	1	no	—
10	C₂×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₄	C₂²	1	1	yes	characteristic
12	SD₁₆	—	2	2	no	—
13	C₂×C₈	—	2	2	no	—
14	C₂×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₈	C₂²	1	1	yes	characteristic
16	C₄⋊C₄	—	2	1	no	—
17	C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄	—	2	1	no	—
20	C₂²⋊C₄	—	2	1	no	—
21	C₂³	—	2	1	no	—
22	C₂³	D₄	1	1	yes	characteristic
23	Q₈	—	2	1	no	—
24	Q₈	—	2	1	no	—
25	D₄	—	2	1	no	—
26	D₄	—	2	1	no	—
27	D₄	—	4	2	no	—
28	C₂×C₄	—	2	1	no	—
29	C₂×C₄	—	2	1	no	—
30	C₂×C₄	—	2	1	no	—
31	C₂×C₄	—	2	1	no	—
32	C₂×C₄	—	2	1	no	—
33	C₂×C₄	D₄	1	1	yes	characteristic
34	C₂×C₄	C₂³	1	1	yes	characteristic
35	C₈	—	2	1	no	—
36	C₈	D₄	1	2	yes	complemented
37	C₂²	—	2	1	no	—
38	C₂²	—	2	1	no	—
39	C₂²	—	4	1	no	—
40	C₂²	—	4	1	no	—
41	C₂²	—	4	1	no	—
42	C₂²	SD₁₆	1	2	yes	complemented
43	C₂²	C₂×D₄	1	1	yes	characteristic
44	C₄	—	2	1	no	—
45	C₄	—	4	1	no	—
46	C₄	—	4	1	no	—
47	C₄	—	4	1	no	—
48	C₄	C₂×D₄	1	1	yes	characteristic
49	C₄	#41	1	1	yes	characteristic
50	C₂	—	2	2	no	—
51	C₂	—	8	1	no	—
52	C₂	#121	1	1	yes	characteristic
53	C₂	C₂×SD₁₆	1	1	yes	characteristic
54	C₂	#135	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³
prime radical	#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×SD₁₆	C₂	semidirect
2. C₂²×C₈	C₂	semidirect
3. C₂²⋊Q₈	C₂	semidirect
5. #106	C₂	semidirect
6. #103	C₂	semidirect
7. #102	C₂	semidirect
36. C₈	D₄	semidirect
42. C₂²	SD₁₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×SD₁₆	2
C₂²×C₈	2
C₂²⋊Q₈	2
#121	2
#106	2
#103	2
#102	2